COMPLEMENTARY AND SUPPLEMENTARY ANGLES

Complementary Angles :

If the sum of two angles is 90⁰, then those two angles are called as complementary angles.

Example :

30° and 60° are complementary angles.

Because,

30° + 60° = 90°

Clearly, 30° is the complement of 60° and 60° is the complement of 30°.

Supplementary Angles :

If the sum of two angles is 180⁰, then those two angles are called as supplementary angles.

Example :

120° and 60° are supplementary angles.

Because,

120° + 60° = 180°

Clearly, 120° is the supplement of 60° and 60° is the supplement of 120°.

Question 1 :

The measure of an angle is 41°. What is the measure of a complementary angle?

Answer :

Let x be the measure of the required complementary angle.

Because x and 41° are complementary angles,

x + 41° = 90°

Subtract 41° from each side.

x = 49°

So, the measure of the complementary angle is 49°.

Question 2 :

The measure of an angle is 62°. What is the measure of a complementary angle?

Answer :

Let x be the measure of the required complementary angle.

Because x and 62° are complementary angles,

x + 62° = 90°

Subtract 62° from each side.

x = 28°

So, the measure of the complementary angle is 28°.

Question 3 :

The measure of an angle is 108°. What is the measure of a supplementary angle?

Answer :

Let x be the measure of the required supplementary angle.

Because x and 108° are supplementary angles,

x + 108° = 180°

Subtract 108° from each side.

x = 72°

So, the measure of the supplementary angle is 72°.

Question 4 :

The measure of an angle is 89°. What is the measure of a supplementary angle?

Answer :

Let x be the measure of the required supplementary angle.

Because x and 41° are supplementary angles,

x + 89° = 180°

Subtract 89° from each side.

x = 91°

So, the measure of the supplementary angle is 91°.

Question 5 :

Two angles are complementary. If one of the angles is double the other angle, find the two angles.

Answer :

Let x be one of the angles.

Then the other angle is 2x.

Because x and 2x are complementary angles, we have

x + 2x = 90°

3x = 90

Divide each side by 3.

x = 30

2x = 2(30) = 60

So, the two angles are 30° and 60°.

Question 6 :

Two angles are complementary. If one angle is two times the sum of other angle and 3, find the two angles.

Answer :

Let x and y be the two angles which are complementary.

x + y = 90° ----(1)

Given : One angle is two times the sum of other angle and 3.

x = 2(y + 3)

x = 2y + 6 ----(2)

Now, substitute (2y + 6) for x in (1).

2y + 6 + y = 90

3y + 6 = 90

Subtract 6 from each side.

3y = 84

Divide each side by 3.

y = 28

Substitute 28 for y in (2).

x = 2(28) + 6

x = 56 + 6

x = 62

So, the two angles are 62° and 28°.

Question 7 :

Find the value of  x :

Answer :

In the picture above, the angles x° and (2x)° are complementary.

x° + (2x)° = 90°

x + 2x = 90

3x = 90

Divide each side by 3.

x = 30

So, the value of x is 30.

Question 8 :

Find the value of  x :

Answer :

In the picture above, the angles

(x + 1)°, (x - 1)° and (x + 3)°

are complementary.

(x + 1)° + (x - 1)° + (x + 3)° = 90°

(x + 1) + (x - 1) + (x + 3) = 90

x + 1 + x - 1 + x + 3 = 90

3x + 3 = 90

Subtract 3 from each side.

3x = 87

Divide each side by 3.

x = 29

So, the value of x is 29.

Question 9 :

Find the value of  x :

Answer :

In the picture above, the angles (2x + 3)° and (x - 6)° are supplementary.

(2x + 3)° + (x - 6)° = 180°

2x + 3 + x - 6 = 180

3x - 3 = 180

Add 3 to each side.

3x = 183

Divide each side by 3.

x = 61

So, the value of x is 61.

Question 10 :

Find the value of  x :

Answer :

In the picture above, the angles

(5x + 4)°, (x - 2)° and (3x + 7)°

are supplementary.

(5x + 4)° + (x - 2)° + (3x + 7)° = 180°

5x + 4 + x -2 + 3x + 7 = 180

9x + 9 = 180

Subtract 9 from each side.

9x = 171

Divide each side by 9.

x = 19

So, the value of x is 19.

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